A187151 Number of walks of length n starting at origin and ending in first quadrant on a square lattice.
1, 2, 8, 26, 108, 382, 1586, 5812, 24044, 89846, 370398, 1401292, 5759826, 21977516, 90111820, 345994216, 1415691244, 5461770406, 22308412934, 86392108636, 352334866238, 1368640564996, 5574504569620, 21708901408216, 88320660937298, 344680279929532, 1400902293406676
Offset: 0
Examples
a(2) = {UU,UR,UD,RU,RR,RL,DU,LR}.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..160 from Vincenzo Librandi)
Programs
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Mathematica
CoefficientList[Series[(Exp[2x]+BesselI[0,2x])^2/4,{x,0,15}],x] * Range[0,15]! Table[2^(-2 + n) (2^n + 2 Hypergeometric2F1[(1 - n)/2, -(n/2), 1, 1] + (2^n Gamma[(1 + n)/2]^2 Mod[n + 1, 2])/(Pi Gamma[1 + n/2]^2)), {n, 0, 30}] (* Benedict W. J. Irwin, Aug 02 2016 *) -
PARI
x='x+O('x^33); Vec(serlaplace((exp(2*x)+besseli(0,2*x))^2/4)) /* Joerg Arndt, Mar 06 2011 */
Formula
E.g.f.: (exp(2*x)+I_0(2*x))^2/4 where I() is the Modified Bessel Function. - Benjamin Phillabaum, Mar 05 2011
Recurrence: (n-1)*n^2*(8*n^3 - 66*n^2 + 171*n - 139)*a(n) = 2*(n-1)^2*(32*n^4 - 288*n^3 + 886*n^2 - 1071*n + 396)*a(n-1) + 24*(2*n-3)*(4*n^4 - 37*n^3 + 114*n^2 - 136*n + 50)*a(n-2) - 32*(n-2)^2*(32*n^4 - 288*n^3 + 886*n^2 - 1071*n + 396)*a(n-3) + 128*(n-3)^2*(2*n-7)*(8*n^3 - 42*n^2 + 63*n - 26)*a(n-4). - Vaclav Kotesovec, Feb 24 2014
a(n) ~ 4^(n-1) * (1 + 2/sqrt(Pi*n)). - Vaclav Kotesovec, Feb 24 2014
From Benedict W. J. Irwin, Aug 02 2016: (Start)
Let b(n) = 2^(2n-2)+2^(n-1)*2F1((1-n)/2,-n/2;1;1).
For odd n, a(n) = b(n),
for even n, a(n) = b(n) + 2^(2n-2)*Gamma((n+1)/2)^2/Gamma(1+n/2)^2/Pi. (End)